The relationship between generalized symmetries and Darboux integrability is well known in the continuous case [1]} and in the discrete case have been investigated in [2]}, [3]}, [4]}. As conjectured in [5]} and proved in [4]},
a quad-equation is Darboux integrable if and only if
it possesses generalized symmetries depending
on arbitrary function in the following form
\(g^{(1)}_{n,m} = R^{(1)}\big (F_{n}\big ( T^{p_{1}}_{n}W_{1},\dots ,T^{q_{1}}_{n}W_{1}\big )\big ), \\g^{(2)}_{n,m} = R^{(2)} \big (G_{m}\big ( T^{p_{2}}_{m}W_{2},\dots ,T^{q_{2}}_{m}W_{2}\big )\big ),\) 
